Theorem ssdif0 1748

Description: Subclass expressed in terms of difference. Exercise 7 of [TakeutiZaring] p. 22.

Assertion

Ref	Expression
ssdif0	⊢ (A ⊆ B ↔ (A ∖ B) = ∅)

Proof of Theorem ssdif0

Step	Hyp	Ref	Expression
1		iman 205	. . . 4 ⊢ ((x ∈ A → x ∈ B) ↔ ¬ (x ∈ A ∧ ¬ x ∈ B))
2		eldif 1496	. . . . 5 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
3	2	negbii 162	. . . 4 ⊢ (¬ x ∈ (A ∖ B) ↔ ¬ (x ∈ A ∧ ¬ x ∈ B))
4	1, 3	bitr4 154	. . 3 ⊢ ((x ∈ A → x ∈ B) ↔ ¬ x ∈ (A ∖ B))
5	4	bial 695	. 2 ⊢ (∀x(x ∈ A → x ∈ B) ↔ ∀x ¬ x ∈ (A ∖ B))
6		dfss2 1497	. 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))
7		eq0 1719	. 2 ⊢ ((A ∖ B) = ∅ ↔ ∀x ¬ x ∈ (A ∖ B))
8	5, 6, 7	3bitr4 158	1 ⊢ (A ⊆ B ↔ (A ∖ B) = ∅)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092   ∖ cdif 1484   ⊆ wss 1487  ∅c0 1707

This theorem is referenced by:  vdif0 1749  pssdifn0 1750  difid 1755  tfi 2244  peano5 2394  tz7.49 2997  oe0m1 3129  php3 3411  strlem1 5691

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708

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